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Argumentation Day 3. Math Bridging Practices June 25, 2014. A Mathematical A rgument. It is… A sequence of statements and reasons given with the aim of demonstrating that a claim is true or false It is not… ( Solely ) an e xplanation of what you did (steps ) - PowerPoint PPT Presentation
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Argumentation Day 3
Math Bridging PracticesJune 25, 2014
A Mathematical Argument
It is…– A sequence of statements and reasons given with
the aim of demonstrating that a claim is true or false It is not…
– (Solely) an explanation of what you did (steps)– A recounting of your problem solving process– Explaining why you personally think it’s true for
reasons that are not necessarily mathematical (e.g., popular consensus; external authority, etc. It’s true because my John said it, and he’s always always right.)
Point of Clarification
What’s 7 + 11?(a) 7 + 11 is 18 because 7 + 1 is 8 and 8 plus10 is 18.This is more retelling steps. (b) 7 + 11 is 18 because 11 is 10 plus 1. I added the 1 onto the 7, to get 8, and then I did 10 plus 8 instead. This is a mathematical argument. It is given to support the claim that 7 + 11 is 18.
Argumentation
Students offer a mathematical reason for why their method is correct
Students offer a logical argument to show how they know that their result is correct
Student work :
When talking about calculations such as:
“I multiplied the cost of one package by 7 because that’s how many packages are needed for 14 days.”
03.16$729.2$
Point of Clarification
Having students generate arguments can happen every day in your class!I would argue it should ha ha ha
What can you make an argument for? Any well formulated claim about something in math that could be determined true or false – no matter how bit or small.
Point of Clarification
“Arguments in math” – need a claim, need evidence, need to know how the evidence shows the claim true (or false).
“Arguments in the courtroom” – need a claim (guilty or not?), need evidence, need to know how the evidence shows the claim true (or false)
“Arguments among friends” “Debates”
Language to help us think about and talk about mathematical arguments
Toulmin’s Model of Argumentation
Claim
Data/Evidence
Warrant
Toulmin’s Model of Argumentation
Claim
Data/Evidence
Warrant
THE ARGUMENT
Toulmin’s Model of Argumentation
Claim7 is an odd number
Data/Evidence2 does not divide 7 evenly
WarrantDefinition of odd/even
An even number is a multiple of 2;
An odd number is not a multiple of 2.
Example
5 and 6 are consecutive numbers, and 5 + 6 = 11 and 11 is an odd number.12 and 13 are consecutive numbers, and 12 + 13 = 25 and 25 is an odd number.1240 and 1241 are consecutive numbers, and 1240 +1241 = 2481 and 2481 is an odd number.That’s how I know that no matter what two consecutive numbers you add, the answer will always be an odd number.
Micah’s Response
Example
5 and 6 are consecutive numbers, and 5 + 6 = 11 and 11 is an odd number.12 and 13 are consecutive numbers, and 12 + 13 = 25 and 25 is an odd number.1240 and 1241 are consecutive numbers, and 1240 +1241 = 2481 and 2481 is an odd number.That’s how I know that no matter what two consecutive numbers you add, the answer will always be an odd number.
Claim
Micah’s Response
Example
5 and 6 are consecutive numbers, and 5 + 6 = 11 and 11 is an odd number.12 and 13 are consecutive numbers, and 12 + 13 = 25 and 25 is an odd number.1240 and 1241 are consecutive numbers, and 1240 +1241 = 2481 and 2481 is an odd number.That’s how I know that no matter what two consecutive numbers you add, the answer will always be an odd number.
Claim
Micah’s Response
Data/Evidence3 examples that fit
the criterion
WarrantBecause if it works
for 3 of them, it will work for all
NOTE: this has the structure of an argument, but this does not show the claim to be true. (not a viable argument)
J: I am a British CitizenB: Prove it
J: I was born in Bermuda
?
Toulmin’s Model of Argumentation
Claim I am a British citizen
Data/EvidenceI was born in Bermuda
WarrantA man born in Bermuda will
legally be a British citizen
Note: What “counts” as a complete or convincing argument varies by grade (age-appropriateness) and by what is “taken-as-shared” in the class (what is understood without stating it and what needs to be explicitly stated). Regardless of this variation, it should be mathematically sound.
Applying Toulmin’s: Ex 1Which is bigger: 73 – 26 or 76 – 26 – 3?
a. 73 – 26 is the same as 76 – 26 – 3. I add 3 to 73 and then take 3 away at the end.
b. 73 – 26 is the same as 76 – 26 – 3. If I add 3 to 73 and then take 3 away at the end, I’ve added nothing overall, so the answer is the same.
c. 73 – 26 is the same as 76 – 26 – 3 because 73 – 26 is 47 and 76 – 26 – 3 is also 47.
Applying Toulmin’s: Ex 1
Which is bigger: 73 – 26 or 76 – 26 – 3?
a. 73 – 26 is the same as 76 – 26 – 3. I can add 3 to 73 and then take 3 away at the end.
b. 73 – 26 is the same as 76 – 26 – 3. If I add 3 to 73 and then take 3 away at the end, I’ve added nothing overall, so the answer is the same.
c. 73 – 26 is the same as 76 – 26 – 3 because 73 – 26 is 47 and 76 – 26 – 3 is also 47.
Data/evidence included; Missing warrant
Warrant included too!
Warrant – I did the math.Not “explanatory”
Applying Toulmin’s: Ex 2Which is bigger? 4 +(x+3)2 or π
a. Pi, because you can’t figure out what 4+(x+3)2 is
b. 4+(x+3)2 because 4 is bigger than pi and (x+3)2 is always positive
c. 4+(x+3)2 because 4 is bigger than pi and (x+3)2 is always positive, so you’re adding a positive value to 4.
GROUP PICTURE in AtriumLUNCH